Chapter 16
Conjugate Heat Transfer in
Ribbed Cylindrical Channels
Armando Gallegos-Muñoz, Nicolás C. Uzárraga-Rodríguez
and Francisco Elizalde-Blancas
Additional information is available at the end of the chapter
http://dx.doi.org/10.5772/49942
1. Introduction
In the last years, many technological advances have emerged in the turbo machinery
industry, mainly in the area of power generation using gas turbines [1]. The main target
in this science field consists on designing and building more efficient machines with a
higher life-time by means of applied research. However, in order to achieve this, it is
necessary that the gas turbine operates at high compression pressure ratios as well as
high turbine inlet temperatures (TIT), but these operating conditions generate thermal
consequences or degradations in the gas turbine components that are exposed to the high
temperatures, like blades and vanes of the first stage. For this reason, it is necessary to
have an internal cooling system in gas turbines to avoid the reduction of the useful life of
their hot components, since the useful life of turbine blades is reduced to half with every
10 – 15 ºC rise in metal temperature [2]. Nowadays basic methods exist, which improve
the gas turbines operating conditions, having as a result improvements of the external
cooling [3], where the use of micro-jets with smaller diameters enhanced the overall heat
transfer coefficient, or internal cooling where square ribbed channels are employed to
study the thermal behaviour of the flow inside the channel [4], turbulence promoters
with different geometries to study the temperature distribution in the gas turbine blades
[5]. Also, it is possible using serpentine passages inside the turbine blade to improve
internal convective cooling [6], ribs in the internal surface of the cooling passages where
the rib-to-rib pitch and angle of attack that yield a maximum heat transfer and maximum
thermal performance are determined [7] or ribs as turbulence promoters to increase the
rate of heat transfer [8]. To increase the heat transfer with minimal friction in compact
heat exchangers, the internal surfaces are ribbed with protuberances that have convex
and concave forms [9]. To study the heat transfer characteristics of laminar flow in
parallel-plate dimpled channels are used [10] or square-channel fitted with baffles [11].
© 2012 Gallegos-Muñoz et al., licensee InTech. This is an open access chapter distributed under the terms
of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/3.0), which permits
unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
470 An Overview of Heat Transfer Phenomena
However, a common way to increase the internal cooling efficiency in gas turbines is to
add ribs, this method offers a better mixed fluid near to the hot internal surface of the
channel thus increasing the thermal heat transfer. The present study shows a numerical
analysis of the first stage blades in a gas turbine with internal cooling system (model
MS7001E) applying the conjugated heat transfer. This method considers the direct
coupling of fluid flow and solid body using the same mesh distribution and numerical
principles for both domains. This coupling is achieved by using boundary conditions
called double-side wall.
2. Mathematical formulation
The numerical analysis of a gas turbine at first stage blade with internal cooling system
considers the solution of the conjugate heat transfer in steady state between the hot
combustion gases flowing around the blade and the coolant flowing inside the cooling
channels of gas turbine blades. The following assumptions are made to model the conjugate
heat transfer problem:
a.
b.
c.
d.
Newtonian fluid,
Compressible and turbulent flow
Rotational frame with relative velocity formulation
Fluid is considered as an ideal gas.
2.1. Governing equations
The applied governing equations are the 3D Reynolds-averaged Navier-Stokes equations
[12], which were solved by commercial Computational Fluid Dynamics software [13]. The
governing equations solved by the model are:
Continuity equation



ρui  ρui  0
xi
(1)
Momentum equation
 

p  τij


 Fi
ρuiu j  ρgi 
x j
xi
x j


(2)
where Fi is the source term which includes contributions due to the body force. Assuming
constant rotational velocity with relative velocity formulation, the source terms due to
rotation are:

  
aco  2ω  ui and ace  ω  ω  r
The term ij is the stress tensor, which is expressed as:
(3)
Conjugate Heat Transfer in Ribbed Cylindrical Channels 471
 u  u 2 u 
j
k   ρuu
 δ
τij  μ  i 
i j
 x j xi 3 ij xk 


(4)
Energy equations
The energy equation for the fluid domain is given by

μ T

 
ρC PuiT  
 ρC P uiT   μφ

C P
xi
xi  Pr xi

(5)
where  is the viscous heating dissipation and ρC P uiT  represents the turbulent heat flux.
The energy conservation equation for the solid is given by the conductive term in the energy
equation. In the solid continuum, only the heat flux due to conduction is included inside
itself. This is described by the heat diffusion equation:

T
 
 λsolid solid   0
xi 
xi 
(6)
where λsolid is the thermal conductivity. Equations (5) and (6) are solved simultaneously by a
conjugate heat transfer analysis. This permits to yield a fully coupled conduction-convection
heat transfer prediction.
Turbulence model
Since the behaviour of the flow in the gas turbine is very chaotic, it is necessary to
incorporate a turbulence model in the numerical analysis to determine the Reynolds
stresses. The Standard k-ε model was used, which relates the Reynolds stresses to the mean
velocity by the Boussinesq hypothesis [13]:
 u  u  2 
u
j
ρuiuj  μt  i 
  ρk  μt k
 x j xi  3 
xk




 δij


(7)
The eddy turbulent viscosity, μt, is calculated by the combination of the turbulent kinetic
energy (k) and the dissipation ratio (ε), as shown by equation (8).
μt  ρ  Cμ 
k2
ε
(8)
where C is a constant. This model offers a reasonable accuracy for a wide range of turbulent
flows in practical engineering problems, in which the turbulent kinetic energy, k, and its
dissipation ratio, ε, are obtained from the following transport equations:
μ  k 

 
 μ  t 
  Gk  Gb  ρε  Ym
ρkui  

xi
x j 
σk  x j 


(9)
472 An Overview of Heat Transfer Phenomena
μ  ε 

 
ε
ε2
 μ  t 
  C1ε  Gk  C3εGb   C2ερ
ρεui  



xi
x j 
σε  x j 
k
k


(10)
In these equations, Gk represents the generation of turbulent kinetic energy due to the mean
velocity gradients, Gb is the generation of turbulent kinetic energy due to buoyancy, and the
quantities σk and σε are the turbulent Prandtl numbers for k and ε, respectively. The
empirical constants appearing in the above equations take the following values: C = 0.09,
C1ε = 1.44, C2ε = 1.92, σk = 1.0 and σε = 1.3 [13].
Equation of state
The density variation in both fluids, the hot combustion gases and cooling air, is assumed
according to the gas ideal law:
ρ
Pop  P
RT
(11)
where R is the gas constant. This equation of state provides the linkage between the energy
equation on one side, and the mass conservation and moment equations on the other. This
linkage emerges from the density variations.
2.2. Computational model and grid
The computational model and mesh were generated in the pre-processor GAMBIT
[14]. In order to avoid many simplifications in the computational model, such as the
use of boundary conditions at surfaces and outlets of the cooling channels, the
computational model was generated using the total blade geometry, which includes the
plenum, gap in the tip, gaps between the seal and the plenum and the ribbed cooling
channels. Figure 1 shows the rotor blade geometry at the first stage of the gas turbine
MS7001E.
In this Figure the internal cooling system can be seen. This system has 13 cylindrical
channels inside the blade. The ribs were placed on the inner surface of the cooling channels
in order to increase the heat transfer from the solid body to the cooling air. Also, in Figure 1
can be seen that the ribs were only placed in the middle blade zone inside the cooling
channels; because in this zone the largest temperature gradient is present [15], causing
failure such as cracks in the blade structure [16].
Three different geometries of rib configurations are studied, which are square, triangular
and semi-circular forms. The ribs are placed perpendicularly to the air flow. Figure 2 shows
a sketch of the different rib parameters used in the square cross-section ribs. These
parameters are used for the other two geometries (triangular and semi-circular). Moreover,
these configurations are applied for both arrangements (full and half-ribs). Figure 3 shows
the form and parameters of the ribs cross-section.
Conjugate Heat Transfer in Ribbed Cylindrical Channels 473
Figure 1. Blade geometry with plenum and its internal features (ribbed cooling channels)
Figure 2. Sketch of different arrays to study the effect of the ribs
Figure 3. Cross-section of the ribs
Due to complexity of the geometry, different computational grid sizes are required. These
grids are not uniform in all directions and were structured by mixing different types of cells
(hexahedral and prismatic elements). For the grid used in the blade computational model
with smooth cooling channels, the mesh density is high in the near-wall region of the blade
474 An Overview of Heat Transfer Phenomena
body. The first wall-adjacent cells height in the vicinity of regions corresponding to the
leading and trailing edge as well as in the suction and pressure side of the blade is 0.0012
mm, while in the region corresponding to the internal cooling channels, the wall-adjacent
cell height is 0.0035 mm. This is developed in order to get a better solution into the
boundary layer, obtaining y+ values closer to unity along the surfaces (ranging from 1 to 5.5).
An analysis to evaluate the grid independence of the numerical solution was developed. The
computational grid has a total of 3809734 mixed cells, 80% of this total corresponding to
fluid (33% air and 47% hot gases) and 20% to solid. Figure 4 shows the grid used in this case.
It can be observed, that the hot combustion gases domain presents a high quantity of
hexahedral elements.
Figure 4. Computational grid for hot combustion gas (red), blade (gray) and cooling air (blue)
domains
For the blade models with ribbed cooling channels, the same grid distribution was used
at the region of hot combustion gases and the external blade surface. The height of the
first wall-adjacent cells in the vicinity of regions corresponding to ribbed cooling
channels was of 0.002 mm, obtaining y+ values less than 5 for all the analyzed rib
configurations. The computational grid used ranges from 6.5 to 7.5 million of mixed
cells. Figure 5 shows the grids used in the internal cooling channels with full-ribs
having an aspect ratio of P/e = 10.
Conjugate Heat Transfer in Ribbed Cylindrical Channels 475
Figure 5. Computational grid for cooling channels domain with full-ribs with P/e = 10. (a) channels and
surfaces, (b) cooling air and (c) ribs
For the models with half-ribs, it was used the same grid distribution showed in the Figure 5,
having a small variation in the ribs domains, defining one half of the domain as fluid and
the other half as solid. Figure 6 shows the grid used in the internal cooling channels with
half-ribs having an aspect ratio of P/e = 10.
Figure 6. Computational grid for cooling channels domain with half-ribs with P/e = 10
2.3. Boundary conditions and thermal properties
The boundary conditions used for the computational model with smooth cooling channels
are defined according to approximate values of the gas turbine operating conditions in
steady state. Figure 7 shows the boundary conditions used in the computational model.
476 An Overview of Heat Transfer Phenomena
Periodicity
Casing surface
(Wall)
Inlet of hot
combustion gases
(Mass flow inlet)
Outlet of mixing
gases-air
(Pressure outlet)
Interfaces
blade-gases
(Wall)
Interfaces
blade-air
(Wall)
Inlet of cooling air
(Mass flow inlet)
Axis of rotation
Figure 7. Boundary conditions in the model of a gas turbine blade
In the hot combustion gases inlet, the operational conditions are: mass flow of 2.47 kg/s,
static pressure of 508700 Pa and total temperature profile is a function of the radial
coordinate. This total temperature profile is described by the next equation [17]:
T r   662195 r 4  2 10 6 r 3  3 10 6 r 2  2 10 6 r  435431
(12)
This total temperature profile is imposed in order to match the oxidation mark of hot
combustion gases over the airfoil of the blade, which has been operating until the end of its
useful life [18], and because this profile is similar to the one obtained in the radial edge
direction of the exit of the transition piece. Figure 8 shows the oxidation marks on the blade.
The units for the independent variables of Equation 12 are: (m) for the rotational radius and
(K) for the total temperature. The turbulence quantities at the inlet of the model are defined
using a turbulence intensity of 5% and 0.006 m for turbulent length scale.
A mass flow of 0.0048 kg/s of air, a static pressure of 897300 Pa, and a total temperature of
853.15 K were specified for each inlet zones of the thirteen cooling channels. Also,
turbulence parameters are defined using a turbulence intensity of 5% and a hydraulic
diameter of 0.004 m for these boundaries. At the left and right sides of the plenum inlets of
cooling air were adjusted with a mass flow of 0.152 kg/s and 0.025 kg/s, respectively. As
well, a turbulence intensity of 5% and turbulent length scale of 0.005 m was set. In the
remaining inlet section of the cooling plenum, boundary conditions were adjusted to the
Conjugate Heat Transfer in Ribbed Cylindrical Channels 477
same conditions used at the inlets of the cooling channels for the parameters of pressure and
temperature. At the exit of the gas-air mixture a static pressure of 473170 Pa, and a backflow
total temperature of 1226 K were specified.
Figure 8. Oxidation (corrosion) marks on the blade [18].
For the solid surfaces the conditions were imposed as no-slip condition, while for the
thermal condition were imposed as coupled. With these considerations it is possible to solve
simultaneously the solid-fluid interfaces. At the interfaces, the temperature and heat flux
could be continuous. These conditions are developed by the use of the boundary conditions
denominated as two-side wall, which can be expressed as:
T fluid  Tsolid
 fluid
T
n
fluid
 solid
(13)
T
n
(14)
solid
Rotational periodic boundary conditions are defined for the suction and pressure side of the
computational model and a nominal angular velocity vector were prescribed.
The flow and heat transfer analysis were performed under the assumption that the fluid
behaviour is compressible and viscous. For the case of the air properties, these are
temperature dependant, while the thermo-physical properties of the solid domain were
assumed to behave like Inconel 738LC alloy. The thermo-physical properties of the fluid and
solid domains are showed in Table 1 [19] and Table 2 [18], respectively.
478 An Overview of Heat Transfer Phenomena
T
[K]
200
300
400
500
600
700
800
900
1000
1100
1200
1300
1400
·106
Cp
[kJ/kg·K]
1.007
1.007
1.014
1.030
1.051
1.075
1.099
1.121
1.141
1.159
1.175
1.189
1.207
λ·103
[W/m·K]
18.1
26.3
33.8
40.7
46.9
52.4
57.3
62.0
66.7
71.5
76.3
82
91
[N·s/m2]
13.25
18.46
23.01
27.01
30.58
33.88
36.98
39.81
42.44
44.90
47.30
49.6
53.0
Cp
[J/kg·K]
420.100
462.110
504.120
525.120
546.130
567.130
588.140
630.150
672.160
714.170
714.170
λ
[W/m·K]
11.83
11.83
11.83
13.70
15.58
17.74
19.76
21.50
23.37
25.39
27.27
ρ
[kg/m3]
8110
8110
8110
8110
8110
8110
8110
8110
8110
8110
8110
Table 1. Fluid properties
T
[K]
294.260
366.480
477.590
588.700
699.810
810.920
922.030
1033.150
1144.260
1255.370
1366.480
Table 2. Properties of Inconel 738LC alloy
2.4. Numerical method
Fluid flow and turbulent heat transfer analysis of the first stage gas turbine blade (MS7001E)
with different ribs configurations placed in the internal cooling channels were realized using
commercial Computational Fluid Dynamic software (FLUENT®). This code allows to solve
the Reynolds averaged Navier-Stokes and the transport equations of the turbulent quantities
for the compressible viscous flow. This CFD code solves the equations using the finite
volume technique [20] to discretize the governing equations inside the computational
domains. The Standard k- ε model [21] was used for all numerical simulations. This model is
a semi-empirical linear eddy viscosity model based on the transport equations for the
Conjugate Heat Transfer in Ribbed Cylindrical Channels 479
turbulent kinetic energy (k) and dissipation rate (ε). The model transport equation for k is
derived from the exact equation, while the model transport equation for ε is obtained using
physical reasoning and little resemblance to its mathematically exact counterpart. The
SIMPLE algorithm was used to link the velocity field and pressure distribution inside the
computational model. This algorithm uses a relation between the velocity and pressure in
order to satisfy the mass conservation, getting a velocity field. The SIMPLE algorithm along
with the implicit time treatment of the flow variables allow to obtain a steady solution or
use rather time steps for unsteady flow computations [13].
The governing equations were solved simultaneously by the approach of the pressure-based
solver. The pressure-based approach is recommended in the literature [13] to be used for
flows with moderate compressibility, since it offers a better convergence. Due to the fact that
the governing equations are non-linear and coupled, several iterations were needed to reach
a converged solution. The Gauss-Seidel linear algorithm was used to solve the set of
algebraic equations obtained by the discretization in FLUENT®. The convergence was
reached when the residuals of the velocity components in the Reynolds averaged NavierStokes equations, continuity and turbulent quantities were smaller than 10-5, while for the
energy conservation equation the residuals were smaller than 10-6.
Six computational equipments were used to solve the model. Each computer has a 3 GHz
processor and 2 GB in RAM. These equipments were connected in a scheme of parallel
processing. Figure 9 shows a sketch of the parallel processing equipment using a basic LAN
topology.
Master
PC
LAN
Network
Switch
Node 1
Figure 9. Sketch of parallel processing.
Node 2
Node 3
Node n
480 An Overview of Heat Transfer Phenomena
3. Results and discussions
In the first part, a comparison between the results obtained experimentally by Kwak [22]
and numerically for the external flow is presented. Also, a comparison between results
obtained from semi-empirical correlations derived from the law of the wall [23] and the
numerical results of the internal flow are presented. In the second part the effects on internal
flow through the internal cooling channels are presented. Finally, the temperature
distribution inside the blade body and the surface temperature distribution in the blade
body with and without ribs are showed.
3.1. Comparison with experimental and semi-empirical correlation data
In order to validate the external flow around the blade a qualitative comparison between
the pressure distribution obtained numerically and experimentally [22], has been
performed. In [22] the pressure distribution on the gas turbine blade of GE-E3 was
measured. Figure 10 shows the comparison between the numerical and experimental
pressure distribution at the middle of the blade. Several turbulence models were used.
The turbulence models used in the comparison were Standard k-ε, RNG k-ε and SST k-ω
models, which are known as two-equation eddy viscosity models (EVMs). The total inlet
pressure and local static pressure around the blade (p0/ps) are plotted as a function of x/Cx,
where x is the axial length measured from the leading edge (LE) of the blade, considered
as the characteristic length.
Figure 10. Pressure distribution around the blade
In Figure 10 can be seen that the pressure distribution, p0/ps, corresponding to numerical
solution on the pressure side (PS) of the blade, agrees with the experimental data. While
some differences for the pressure distribution, p0/ps, on the suction side (SS) of the blade, are
observed.
Conjugate Heat Transfer in Ribbed Cylindrical Channels 481
For comparison purposes of the internal flow in the cooling channels, the pressure drop given by
the section with square full-ribs having an aspect ratio of P/e = 10 for the central cooling channel
was determined. This pressure drop was calculated by the friction factor for ribbed tubes, using
semi-empirical correlations derived from the law of wall. Nikuradse [23] developed a friction
factor correlation to be used in geometries with sand-grain roughness. His results were excellent
for a wide range of roughness sizes. This correlation is expressed by Equation (15).
 
R e   2 f  2.5 ln  2e Dh   3.75
(15)
The term e+ = eu*/v is the roughness Reynolds number and Dh is the hydraulic diameter
which can be expressed as:
Dh 
4A
Pw
(16)
where A is the cross section area and Pw is the wetted perimeter of the cooling channel.
Webb [24] used Equation (15) to calculate frictional data for turbulent flows in ribbed tubes
with circular cross-section, obtaining excellent results. They found that the roughness
function could be correlated as Equation (17).
 
R e   0.95  P e 
0.53
(17)
This equation is valid in the range e+ > 35. By solving Equations (15) and (17) the friction
factor can be found from the geometrical parameters of the internal structure of the ribbed
channels. Equation (18) shows the result obtained.
f 
2


 R  2.5 ln  2e Dh   3.75 


(18)
2
Equation (18) is valid for channels with ribs placed 90º to the flow direction and an aspect
ratio of P/e = 10. Table 3 shows the pressure drop calculated by the Equation (18) and the
numerical results obtained in this work.
Δp (kPa)
Correlation Eq. (18)
3893.90
Numeric
3917.20
Table 3. Pressure drop comparison between analytical and numerical results
The pressure drop calculated numerically presents a good approximation, having an
absolute difference of 3.25%.
3.2. Effects of the internal flow through internal cooling channels
In order to determine the effects generated by the ribs, a line along the central cooling
channel was created. This centerline is dimensionless with a range from 0 to 1, where y is the
482 An Overview of Heat Transfer Phenomena
dimensionless distance of the axial length of the internal cooling channel, measured from
the base of the blade until the outlet of the cooling channel. This centerline was used to
obtain data of the flow parameters, such as temperature, velocity, Mach number and
pressure loss.
Temperature distribution
Figure 11 shows the temperature distributions of the coolant flow along the centerline of the
central cooling channels. Figures 11a and 11c show the results for different types of full-ribs,
and Figures 11b, and 11d show the results for the half-ribs studies, having a ratio of P/e = 10
and P/e = 20, respectively. The higher and smaller temperatures are presented for the
configurations of full-ribs in the ribbed section (Figs. 11a and 11c.), reaching temperatures
from 937 K to 741 K, respectively. While the half-ribs configurations offer a smaller difference
of temperature in this section. This range is between temperature values of 927 K and 791 K.
As can be seen in Figure 11, the temperature presents a moderate increase at the smooth inlet
section. In the ribbed section, the temperature presents a periodic variation, due to the
acceleration and deceleration of the flow inside the cooling channels caused by the presence of
the ribs. At the end of the ribbed section, the temperature strongly decreases. This effect is
similar to compressible flow with heat transfer (Rayleigh curve) [25], where the temperature
decreases to reach a Mach number larger than one. In the smooth outlet section, the
temperature increases at the beginning of the section, due to the decrement in the Mach
number. The temperature suffers a decrement while the flow gets closer to the outlet channel.
The temperature contours along the central cooling channel at a longitudinal plane are shown
in Figure 12. Figures 12a, 12c and 12e show the results for different types of full-ribs, and
Figures 12b, 12d and 12f show results for the half-ribs, both models have a ratio of P/e = 10. The
fluid temperature increases while it goes along the channel for all cases (arrow indicates the
direction of flow). The surface temperature of the channels is higher in the cases with half-ribs.
Thus, the flow is heated at surface near the ribs. For the cases with full-ribs, the surface
temperature is lower, showing a more uniform temperature distribution near to the wall.
The triangular ribs configuration presents the lowest temperatures, because this
configuration offers the best cooling design inside the blade body. These effects are similar
to the configurations with ribs whose P/e = 20.
Velocity and mach number distribution
Figure 13 shows the comparison between velocity magnitude and Mach number
distributions obtained for the cases of blades with smooth and ribbed channels.
In the ribbed channel corresponding to the configuration of square full-ribs with an aspect
ratio of P/e = 10, the highest Mach number and velocity are obtained, whose values are 1.45
and 823 m/s, respectively. The smooth channel presents a continuous acceleration of the
flow through the channels. In the case of the ribbed channel, the flow is moderately
accelerated in the smooth inlet section. The flow becomes unstable in the ribbed section;
experiencing acceleration and deceleration continuously. At the end of this section, in the
last rib, the flow is strongly accelerated as can be seen in Figure 13. High velocities decrease
Conjugate Heat Transfer in Ribbed Cylindrical Channels 483
at the beginning of smooth outlet section, and then accelerate again towards the outlet of the
smooth channel. This effect is due to the rotational force.
Smooth inlet section
950
Ribbed section
Smooth outlet section
Temperature [K]
900
850
800
Smooth
Square full-ribs P/e=10
Triangular full-ribs P/e=10
Semi-circular full-ribs P/e=10
750
700
0.0
0.2
0.4
Smooth inlet section
950
y/ymax [-]
(a)
0.6
Ribbed section
0.8
1.0
Smooth outlet section
Temperature [K]
900
850
800
Smooth
Square half-ribs P/e=10
Triangular half-ribs P/e=10
Semi-circular half-ribs P/e=10
750
700
0.0
0.2
0.4
y/ymax [-]
(b)
0.6
0.8
1.0
484 An Overview of Heat Transfer Phenomena
Ribbed section
Smooth inlet section
950
Smooth outlet section
Temperature [K]
900
850
800
Smooth
Square full-ribs P/e=20
Triangular full-ribs P/e=20
Semi-circular full-ribs P/e=20
750
700
0.0
0.2
0.4
Smooth inlet section
950
y/ymax [-]
(c)
0.6
Ribbed section
0.8
1.0
Smooth outlet section
Temperature [K]
900
850
800
Smooth
Square half-ribs P/e=20
Triangular half-ribs P/e=20
Semi-circular half-ribs P/e=20
750
700
0.0
0.2
0.4
y/ymax [-]
0.6
0.8
1.0
(d)
Figure 11. Temperature distribution at the centerline inside of the central cooling channel
Conjugate Heat Transfer in Ribbed Cylindrical Channels 485
Figure 12. Temperature Contours [K] along the central cooling channel for the different types of ribs
with P/e = 10
900Smooth inlet section
Smooth outlet section
1.52
Smooth channel
Ribbed channel
800
1.32
Mach númber [-]
Velocity [m/s]
Ribbed section
700
1.12
600
0.92
500
0.72
400
0.52
300
0.0
0.2
0.4
y/ymax
0.6
0.8
1.0
Figure 13. Velocity magnitude [m/s] and Mach number distributions along the centreline inside of the
central cooling channels.
486 An Overview of Heat Transfer Phenomena
In order to have a better description of the effects caused by the acceleration and
deceleration of the flow mentioned above, Figure 14 shows the contours of the axial velocity
through the central cooling channel at a longitudinal plane. Figures 14a, 14c and 14e show
the results for the different types of full-ribs. Figures 14b, 14d and 14f show the results for
the half-ribs, both models have a ratio of P/e = 10. As it can be observed, the flow is strongly
accelerated every time that it passes between the rib tips, which provoke that the fluid
increases its velocity due to area reduction. Then, the area increases again to decelerate the
fluid flow. These fluctuations of acceleration are presented periodically in the cooling
channel, generating variations in the flow parameters. This effect is more noticeable in the
cases of half-ribs, having the higher bulk velocity in the channel at each rib. Also, the higher
velocity is extended downstream of the ribs tip, where the flow follows a wavy path in the
bulk section of the channel. Also, it is observed that in the half-rib cases, the flow is forced
towards the surface of the opposite rib.
On the other hand, the square and semi-circular ribs produce recirculation zones as well as
stagnation points over the upstream and downstream rib surfaces, respectively. The
triangular ribs only produce recirculation zones in the downstream surfaces. These effects
are similar to configurations with a ratio of P/e = 20.
Figure 14. Contours of axial velocity [m/s] along to central cooling channel for the different types of
ribs with P/e = 10
Conjugate Heat Transfer in Ribbed Cylindrical Channels 487
Pressure loss
The local static pressure is presented in terms of the normalized pressure difference,
calculated by the equation (19)
p 
p  pexit
(19)
0.5u 2
where p is the local static pressure, pexit is the pressure at the outlet of the cooling channel,
where the cooling air is mixed with the hot combustion gases and the average velocity u is
calculated by the channels mass flow rate. Figures 15 and 16 show the local normalized
static pressure distribution for the different rib configurations with an aspect ratio, P/e, of 10
and 20, respectively.
5.0
Smooth inlet section
Ribbed section
4.5
Pressure drop (Δp) [-]
4.0
3.5
3.0
Smooth outlet section
Smooth
Square full-ribs P/e=10
Triangular full-ribs P/e=10
Semi-circular full-ribs P/e=10
Square half-ribs P/e=10
Triangular half-ribs P/e=10
Semi-circular half-ribs P/e=10
2.5
2.0
1.5
1.0
0.5
0.0
-0.5
0.0
0.2
0.4
y/ymax [-]
0.6
0.8
1.0
Figure 15. Static pressure distribution along the central cooling channel for different types of ribs with
P/e = 10
In Figures 15 and 16 can be observed that the slope of pressure drop in the smooth inlet
section decreases due to a gradual reduction of channel cross-section. This area reduction
is localized in the joint between the plenum and the blade. After this section, the pressure
increases while the channel distance increases to the ribbed section. This is produced by
the stagnation point when the flow shocks with the first ribs. In the ribbed section,
the slope of the pressure drop becomes unstable, presenting periodical increments and
decrements due to the cross-section variation, producing accelerations and decelerations
of the flow. At the smooth outlet section, the slope of the pressure drop is relatively
higher than that in the smooth inlet section. This is due to the increase of the flow velocity
at this zone due to the rotational force, ejecting the flow inside the hot gases stream in the
tip of the blade.
488 An Overview of Heat Transfer Phenomena
5.0
Smooth inlet section
Smooth outlet section
Ribbed section
Smooth
Square full-ribs P/e=20
Triangular full-ribs P/e=20
Semi-circular full-ribs P/e=20
Square half-ribs P/e=20
Triangular half-ribs P/e=20
Semi-circular half-ribs P/e=20
4.5
Pressure drop (Δp) [-]
4.0
3.5
3.0
2.5
2.0
1.5
1.0
0.5
0.0
-0.5
0.0
0.2
0.4
y/ymax [-]
0.6
0.8
1.0
Figure 16. Static pressure distribution along the central cooling channel for different types of ribs with
P/e = 20
3.3. Temperature distribution inside the blade body
Figure 17 shows the temperature contours at a transversal plane of the blade body right in
the middle of the blade for the cases with smooth and full-rib channels with a P/e = 10.
Comparing the results obtained, it is possible to observe an improvement in the blade
internal cooling, allowing a decrement in the internal surface temperature of the cooling
channels. Thus, the heat removed by the coolant is increased.
As can be seen in Figure 17, the maximum temperature decreases, approximately about 10
to 20 degrees and is reached close to the internal surfaces for the cases of blade with ribbed
channels (Figures 17b, 17c and 17d), noticing that the cooling zone covers the major part of
the internal cooling channels, propagating to the leading and trailing edge. In the cases of
the blade with smooth channels, it is only present a smaller cooling zone at the three central
cooling channels (Figure 17a).
Models with square and triangular cross-section full-ribs show a similar temperature
distribution and major heat dissipation compared with the semi-circular full-ribs.
Mazur [16] performed an analysis of a gas turbine bucket failure made of Inconel 738LC
super alloy. This bucket operated for 24,000 hours. Mazur et al. [16] found that the
maximum stresses are present in the blade cooling channels, producing cracks. Figure 18
shows that kind of cracks. These start in the coating of the cooling channels, propagating
and following intergranular trajectories, reaching a depth up to 0.4 mm.
Conjugate Heat Transfer in Ribbed Cylindrical Channels 489
Figure 17. Temperature contours [K] in the metal, a) smooth, b) square full-ribs, c) triangular full-ribs
and d) semi-circular full-ribs, cooling channels
Figure 18. Cracks in the central cooling channels [16]
490 An Overview of Heat Transfer Phenomena
In this way, the effect generated by increasing the internal cooling zone produces an
increment in the useful life of the blade, since the useful life of gas turbine blades is reduced
to half with every 10-15 °C rise in metal temperature [2]. On the other hand, the use of ribs
increases the heat transfer, generating an increase in thermal gradients at internal surface of
cooling channels. In the leading edge another interesting effect is presented. There is a
minor penetration of the blade body temperature (Figures 17b to 17d) caused by the use of
the ribs. However, it cannot be adequate due to the fact that the thermal gradients at the
leading edge are increasing. Due to these thermal effects, these zones must be taken into
account to be protected by means of the film cooling method.
Figure 19 shows the blade profile right in the middle along the blade height. In this section,
a perpendicular line to the blade chord is created to obtain the temperature distribution
inside the solid body as well as in the cooling air through the central cooling channel. The
distance is a dimensionless parameter, taking values between 0 and 1, starting in the suction
side and ending in the pressure side, respectively.
Figure 19. Perpendicular line to the blade chord where data is obtained
In Figures 20a and 20b the temperature distributions for the cases with full and half-ribs are
presented, respectively. Both models have an aspect ratio between pitch and height of the
ribs (P/e) of 10. Figure 20a shows that the cases of ribbed channels present a significant
decrement in the temperatures inside the solid body in comparison with smooth channels.
The triangular full-ribs model presents a higher temperature decrement, reaching a
temperature decrement up to 20 K closer to the channel surface and 10 K in the pressure and
suction sides. Figure 20b shows that square and semi-circular half-ribs do not offer a
considerable decrement in the temperature inside the blade body, since the temperature
distributions are very similar to the case with smooth cooling channels. The case with
triangular half-ribs presents a smaller temperature decrement than the case with triangular
full-ribs (Figure 20a). The decrement of temperature achieved by this configuration is
between 10 K near to the channel surface and 4 K in the pressure and suction sides. In the
Conjugate Heat Transfer in Ribbed Cylindrical Channels 491
fluid section, the temperature is bigger in all the ribbed cases than in the smooth case,
obtaining improved heat dissipation to the cooling air.
Pressure side
Fluid
Suction side
1150
Temperature [K]
1100
1050
1000
950
Smooth
Square full-ribs P/e=10
Triangular full-ribs P/e=10
Semi-circular full-ribs P/e=10
900
850
800
0.0
0.2
0.4
z/zmax [-]
(a)
Suction side
1150
0.6
Fluid
0.8
1.0
Pressure side
1100
Temperature [K]
1050
1000
950
Smooth
Square half-ribs P/e=10
Triangular half-ribs P/e=10
Semi-circular half-ribs P/e=10
900
850
800
0.0
0.2
0.4
z/zmax [-]
0.6
0.8
1.0
(b)
Figure 20. Temperature distributions in the blade with ribbed cooling channels with P/e = 10
Figures 21a and 21b show the temperature distributions for the cases of cooling channels
with full and half-ribs with an aspect ratio of P/e = 20. Figure 21a shows that the temperature
distributions in the solid body are similar for the three ribbed cases, having the lowest
temperature in the square ribs model. With these configurations a higher penetration of the
cooling blade using any rib geometry is achieved. These configurations present a similar
behaviour, in comparison with the results presented in Figure 20a. These cases present a
492 An Overview of Heat Transfer Phenomena
temperature reduction up to 22 K in regions close to the channel surface and up to 10 K in
the pressure and suction sides. In Figure 21b can be observed a uniform behaviour of the
temperature distribution for the three types of ribs. This behaviour is basically the same for
all the cases. However, the blade cooling is affected due to the temperature distributions
obtained for all the cases with different tendency to be similar for the smooth case, having a
smaller improvement on the blade temperature when compared with the temperature
profiles shown in Figure 20b.
Fluid
Suction side
1150
Pressure side
Temperature [K]
1100
1050
1000
950
900
Smooth
Square full-ribs P/e=20
Triangular full-ribs P/e=20
Semi-circular full-ribs P/e=20
850
800
0.0
0.2
0.4
z/zmax [-]
0.6
0.8
1.0
(a)
Fluid
Suction side
1150
Pressure side
Temperature [K]
1100
1050
1000
950
900
Smooth
Square half-ribs P/e=20
Triangular half-ribs P/e=20
Semi-circular half-ribs P/e=20
850
800
0.0
0.2
0.4
z/zmax [-]
0.6
0.8
1.0
(b)
Figure 21. Temperature distributions in the blade with ribbed cooling channels with P/e = 20
Conjugate Heat Transfer in Ribbed Cylindrical Channels 493
The temperature distributions of cooling air presented in Figures 20 and 21 have a
symmetrical parabolic behaviour due to mixing flow. This is generated by placing ribs,
while the profile related with the smooth channel has an asymmetrical behaviour. In this
case the profile presents a tendency to attach to the pressure side due to the blade rotational
force.
The effect of having a symmetric profile improves the heat transfer from the internal surface
of the cooling channels to the air, due to the turbulence generated in the flow which is
increased because of the ribs removing a high quantity of heat.
Figure 22. Surface temperature distributions [K] on the blade with a) smooth cooling channel and b)
ribbed cooling channels with P/e = 10
494 An Overview of Heat Transfer Phenomena
3.4. Surface temperature distribution in the body blade
Figure 22 shows a comparison between the surface temperature distribution on the pressure
side and the suction side of the blade for the cases of models with smooth and ribbed
cooling channels. The temperature distributions of the blade with ribbed internal cooling
channels correspond to the configuration with square full-ribs with a ratio P/e = 10. As can
be seen in Figure 22, internal cooling generated by ribs has an effect on the blade surface
temperature, since it presented a substantial decrease in surface temperature on the pressure
and suction sides of the blade. Also, it is observed a reduction of the spot of maximum
temperature on the leading and trailing edges of the blade generated by the parabolic
temperature profile at the inlet of the hot combustion gases. Another effect which can be
seen is the stain of cooling at the root of the blade on the suction side, which is generated by
the flow of air entering the vane platform.
4. Conclusions
In the present work, a numerical study was performed, with the aim to assess the effect
generated by the ribs in the temperature distributions inside the blade body as well as the
pressure drop through the cooling channels with different types of ribs. The main
conclusions are:
The validation of the numerical model by comparing the internal and external flow with
experimental [22] and semi-empirical [23, 24] data was developed. The pressure drop in the
internal flow obtained through the numerical solution and semi-empirical [23, 24] data,
offers a close enough agreement, with an absolute difference of 3.25%.
The higher and smaller temperatures of the internal flow are presented for the
configurations of full-ribs in the ribbed section, reaching temperatures from 937 K to 741 K,
respectively, while the half-ribs configurations offer a smaller difference of temperature in
this section. This range is between temperature of 927 K and 791 K. The configurations of
full-ribs have a larger contact area than configurations of half-ribs. Due to this reason, the
configurations of full-ribs remove more thermal energy from the blade body.
The configuration with full-ribs with P/e = 20, offers the best cooling with any rib type. This
could be due to the fact that the flow has a high recirculation zone between the ribs,
generating a hydrodynamic perturbation to provoke a separation of the boundary layer.
The acceleration and deceleration effects, which are presented in the ribbed section, play an
important role in the flow behaviour of the compressible fluid, since the high velocity of the
flow shows a strong influence on the variations of temperature in the flow field. The highest
Mach number and velocity are obtained with the ribbed channel, whose values are 1.45 and
823 m/s, respectively, while that smooth channel presents a continuous acceleration of the
flow along the channels.
The ratio between the required inlet pressure in the cooling channels and the outlet pressure
increases from 4 to 4.5 times approximately for the cases with full-ribs with aspect ratios
Conjugate Heat Transfer in Ribbed Cylindrical Channels 495
(P/e) of 10 and 20, respectively. For the half-ribs, this ratio is between 3 to 4 times,
approximately. These values are higher than the values obtained in smooth cooling
channels.
The ribbed cooling channels present different pressure drops, ordered from higher to lower
pressure drops, they are triangular, square and semi-circular ribs, respectively. The
triangular ribs offer the highest cooling effects of the analyzed cases; however, this
configuration presents the highest pressure drop when compared with any other case.
The turbulence promoters allow to obtain a maximum temperature decrease, approximately
about 10 to 20 degrees, close to the internal surfaces of the blade body. This allows reducing
damages by fatigue and thermal stresses.
Author details
Armando Gallegos-Muñoz, Nicolás C. Uzárraga-Rodríguez and Francisco Elizalde-Blancas
Department of Mechanical Engineering, University of Guanajuato, Salamanca, Gto., Mexico
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